Template:Langle/doc

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This is the left-handed angular bracket used for writing averages or bra-ket notation, with other applications primarily in mathematics and physics, for use when inline html rendering is desired rather than Template:TeX rendering.

Examples

There is no need to use spaces around the template, due the large spacing in the template glyph.

Examples of bras

The superposition of states can be written ⟨p| +⟨q| +⟨χ| +⟨ψ|, which is inline with the text.

Another superposition of states: ⟨P| +⟨Q| +⟨Φ| +⟨Ψ|, again inline.

The superposition of states can be written {{langle}}p| +{{langle}}q| +{{langle}}χ| +{{langle}}ψ|, which is inline with the text.

Another superposition of states: {{langle}}P| +{{langle}}Q| +{{langle}}Φ| +{{langle}}Ψ|, again inline.
In conjunction with {{Rangle}}

One sum of inner products is ⟨p|qTemplate:Rangle+⟨χ|ψTemplate:Rangle, a real number.

Another sum of inner products is ⟨P|QTemplate:Rangle+⟨Φ|ΨTemplate:Rangle, another real number.

{{langle}}p|q{{rangle}}+{{langle}}χ|ψ{{rangle}}

{{langle}}P|Q{{rangle}}+{{langle}}Φ|Ψ{{rangle}}

The average of a quantity q may be written ⟨qTemplate:Rangle. The root mean square is then √⟨q2Template:Rangle, i.e. square every value, then average, then take the root.

The average of a quantity ''q'' may be written {{langle}}''q''{{rangle}}. The root mean square is 
then √{{langle}}''q''<sup>2</sup>{{rangle}}, i.e. square every value, then average, then take the root.

See also

For use with {{Rangle}}